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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Relations between Graded EquipollenceAnd Fuzzy C-measures Of Finite Fuzzy Sets
Michal Holcapek
Centre of Excellence IT4Innovationsdivision of the University of Ostrava
Institute for Research and Applications of Fuzzy ModelingOstrava 1, Czech Republic
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Outline1 Motivation and preliminaries
Cardinal theory (finite case)Fuzzy cardinal theory (FCT): a surveyFuzzy sets and fuzzy cardinals
2 Graded equipollence of fuzzy sets3 Fuzzy c-measures of finite fuzzy sets
MotivationAxiomatic definitionExpression of fuzzy c-measuresExample
4 Relation between graded equipollence and c-measuresof finite fuzzy sets
Preliminary notionsOne-to-one mappings vs. equivalence of fuzzy cardinals
5 Conclusion
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
A poor interest about fuzzy cardinal theory
S. GottwaldFuzzy uniqueness of fuzzy mappings. Fuzzy Sets andSystems, 3:49–74, 1980.
L. ZadehA computational approach to fuzzy quantifiers in naturallanguages. Comp. Math. with Applications 9 (1983) 149–184
M. WygralakVaguely defined objects. Representations, fuzzy sets andnonclassical cardinality theory. Theory and Decision Library.Kluwer Academic Publisher, 1996.
M. WygralakCardinalities of Fuzzy Sets. Kluwer Academic Publisher, Berlin,2003.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Cardinal theory (finite case)
Example
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∗
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�
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Apples Pears
How to compare the mass of apples and pears?
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Cardinal theory (finite case)
Functional approach to compare the size of sets
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∗
∗
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�
��
��
Apples Pears
f
Using a one-to-one correspondence (functional approach).
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Cardinal theory (finite case)
Approach based on ordinal numbers
∗∗
∗
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�
��
��0
1
2
3
4
Apples Pears5
Using ordinal (cardinal) numbers.
|Apples| = 5 = |Pears|
Von Neumann construction of natural numbers:
0 = ∅, 1 = 0 ∪ {0}, . . . , 5 = 4 ∪ {4}, . . .
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy cardinal theory (FCT): a survey
Two directions in the fuzzy cardinal theory
We can distinguish the approaches based on1 the relation “to have the same fuzzy cardinality”
|A| = |B| or |A| ∼ |B| = α (graded approach)
2 fuzzy measures similar to the cardinality measure
C(A) = real number or C(A) = fuzzy number
i.e.
C : Ffin→ N
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy cardinal theory (FCT): a survey
Two directions in the fuzzy cardinal theory
We can distinguish the approaches based on1 the relation “to have the same fuzzy cardinality”
|A| = |B| or |A| ∼ |B| = α (graded approach)
2 fuzzy measures similar to the cardinality measure
C(A) = real number or C(A) = fuzzy number
i.e.
C : Ffin→ N
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy cardinal theory (FCT): a survey
Two directions in the fuzzy cardinal theory
We can distinguish the approaches based on1 the relation “to have the same fuzzy cardinality”
|A| = |B| or |A| ∼ |B| = α (graded approach)
2 fuzzy measures similar to the cardinality measure
C(A) = real number or C(A) = fuzzy number
i.e.
C : Ffin→ N
logo
Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy cardinal theory (FCT): a survey
Two directions in the fuzzy cardinal theory
We can distinguish the approaches based on1 the relation “to have the same fuzzy cardinality”
|A| = |B| or |A| ∼ |B| = α (graded approach)
2 fuzzy measures similar to the cardinality measure
C(A) = real number or C(A) = fuzzy number
i.e.
C : Ffin→ N
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy cardinal theory (FCT): a survey
Several natural questions about ∼ andC : Ffin→ N.
One can askWhat structure of truth values is suitable? (residuatedlattice, MV-algebra, IMTL-algebra???)What is Ffin? (a set or class of fuzzy sets???)What is N? (set or class of finite fuzzy cardinals???)How to establish the degree to which two (finite) fuzzy setshave the same cardinality (using one-to-onecorrespondences between fuzzy sets, or α-cuts???).What properties have to keep the mapping C to besomething like the cardinality measure? (additive measure,cardinality measure for the classical set???)
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Residuated-dually residuated lattice
Łukasiewicz algebraAn algebra ([0,1],∧,∨,⊗,→,⊕,) is the Łukasiewicz algebra,if for a,b, c ∈ [0,1], we have
¬a = 1− a,a⊗ b = max(a + b − 1,0),a⊕ b = min(a + b,1) (dual operation to ⊗),a→ b = min(1− a + b,1),a b = max(a− b,0) (dual operation to→).
Common denotationWe use � ∈ {∧,⊗} and � ∈ {∨,⊕}.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Fuzzy sets in the universe of countable setsCount
DefinitionA mapping A : x → L is called a countable fuzzy set in Count, ifx is a set in Count. The class of all countable fuzzy sets inCount is denoted by Fcount.
Definition∅ : ∅ → L is the empty fuzzy set,if Dom(A) contains only one element, then A is a singleton,Supp(A) = {x ∈ Dom(A) | A(x) > ⊥} is a support of A,A is a finite fuzzy set, if Supp(A) is a finite set,Ffin denotes the class of all finite fuzzy sets in Count.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Fuzzy sets in the universe of countable setsCount
DefinitionA mapping A : x → L is called a countable fuzzy set in Count, ifx is a set in Count. The class of all countable fuzzy sets inCount is denoted by Fcount.
Definition∅ : ∅ → L is the empty fuzzy set,if Dom(A) contains only one element, then A is a singleton,Supp(A) = {x ∈ Dom(A) | A(x) > ⊥} is a support of A,A is a finite fuzzy set, if Supp(A) is a finite set,Ffin denotes the class of all finite fuzzy sets in Count.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Fuzzy sets in the universe of countable setsCount
DefinitionA mapping A : x → L is called a countable fuzzy set in Count, ifx is a set in Count. The class of all countable fuzzy sets inCount is denoted by Fcount.
Definition∅ : ∅ → L is the empty fuzzy set,if Dom(A) contains only one element, then A is a singleton,Supp(A) = {x ∈ Dom(A) | A(x) > ⊥} is a support of A,A is a finite fuzzy set, if Supp(A) is a finite set,Ffin denotes the class of all finite fuzzy sets in Count.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Fuzzy sets in the universe of countable setsCount
DefinitionA mapping A : x → L is called a countable fuzzy set in Count, ifx is a set in Count. The class of all countable fuzzy sets inCount is denoted by Fcount.
Definition∅ : ∅ → L is the empty fuzzy set,if Dom(A) contains only one element, then A is a singleton,Supp(A) = {x ∈ Dom(A) | A(x) > ⊥} is a support of A,A is a finite fuzzy set, if Supp(A) is a finite set,Ffin denotes the class of all finite fuzzy sets in Count.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Fuzzy sets in the universe of countable setsCount
DefinitionA mapping A : x → L is called a countable fuzzy set in Count, ifx is a set in Count. The class of all countable fuzzy sets inCount is denoted by Fcount.
Definition∅ : ∅ → L is the empty fuzzy set,if Dom(A) contains only one element, then A is a singleton,Supp(A) = {x ∈ Dom(A) | A(x) > ⊥} is a support of A,A is a finite fuzzy set, if Supp(A) is a finite set,Ffin denotes the class of all finite fuzzy sets in Count.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Fuzzy sets in the universe of countable setsCount
DefinitionA mapping A : x → L is called a countable fuzzy set in Count, ifx is a set in Count. The class of all countable fuzzy sets inCount is denoted by Fcount.
Definition∅ : ∅ → L is the empty fuzzy set,if Dom(A) contains only one element, then A is a singleton,Supp(A) = {x ∈ Dom(A) | A(x) > ⊥} is a support of A,A is a finite fuzzy set, if Supp(A) is a finite set,Ffin denotes the class of all finite fuzzy sets in Count.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Equivalence relation for fuzzy sets
DefinitionWe shall say that fuzzy sets A and B are the equivalent fuzzysets (symbolically, A ≡ B), if Supp(A) = Supp(B) andA(x) = B(x) for any x ∈ Supp(A).
Definitioncls(A) denotes the class of all equivalent fuzzy sets with A.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Equivalence relation for fuzzy sets
DefinitionWe shall say that fuzzy sets A and B are the equivalent fuzzysets (symbolically, A ≡ B), if Supp(A) = Supp(B) andA(x) = B(x) for any x ∈ Supp(A).
Definitioncls(A) denotes the class of all equivalent fuzzy sets with A.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Operations in Fcount
DefinitionLet A,B ∈ Fcount, x = Dom(A) ∪ Dom(B) and A′ ≡ A, B′ ≡ Bsuch that Dom(A′) = Dom(B′) = x. Then
the union of A and B is a mapping A∪B : x → L defined by
(A ∪ B)(a) = A′(a) ∨ B′(a),
the intersection of A and B is a mapping A ∩ B : x → Ldefined by
(A ∩ B)(a) = A′(a) ∧ B′(a),
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Operations in Fcount
DefinitionLet A,B ∈ Fcount, x = Dom(A) ∪ Dom(B) and A′ ≡ A, B′ ≡ Bsuch that Dom(A′) = Dom(B′) = x. Then
the union of A and B is a mapping A∪B : x → L defined by
(A ∪ B)(a) = A′(a) ∨ B′(a),
the intersection of A and B is a mapping A ∩ B : x → Ldefined by
(A ∩ B)(a) = A′(a) ∧ B′(a),
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Example
Consider the Łukasiewicz algebra LFor A = {1/a,0.4/b} and B = {0.6/a,0.2/c} we have
A ∪ B = {1/a,0.4/b,0.2/c},A ∩ B = {0.6/a,0/b,0/c},
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Generalized cardinals in FCT for finite fuzzysets
DefinitionA generalized cardinal A (over N) is an �-convex fuzzy setA : N→ L, i.e.
A(i)� A(j) ≤ A(k), i ≤ k ≤ j .
N denotes the set of all generalized cardinals.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Structure of fuzzy cardinals
Addition of fuzzy cardinals and neutral element (zeroelement)
(A + B)(i) =∨
k ,l∈Nk+l=i
(A(k)� B(l)),
0(k) =
{1, k=0;0, otherwise.
TheoremThe triplet (N,+,0) is a commutative monoid.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Motivation and preliminaries
Fuzzy sets and fuzzy cardinals
Structure of fuzzy cardinals
Addition of fuzzy cardinals and neutral element (zeroelement)
(A + B)(i) =∨
k ,l∈Nk+l=i
(A(k)� B(l)),
0(k) =
{1, k=0;0, otherwise.
TheoremThe triplet (N,+,0) is a commutative monoid.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Graded equipollence of fuzzy sets
How to define degrees of one-to-one mappings
DefinitionLet A,B ∈ Ffin, x , y ∈ Count and f : x → y be a one-to-onemapping of x onto y in Count. We shall say that f is aone-to-one mapping of A onto B in the degree α with respect to�, if Supp(A) ⊆ x ⊆ Dom(A) and Supp(B) ⊆ y ⊆ Dom(B) and
α =⊙z∈x
(A(z)↔ B(f (z))).
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Graded equipollence of fuzzy sets
How does it work?
Dom(A)
Dom(B)
x
y
f
Supp(A)
Supp(B)
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Graded equipollence of fuzzy sets
But we can imagine much more!
Dom(A)
Dom(B)
x
y = Dom(B′)
f
Supp(A)
Supp(B) = Supp(B′)
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Graded equipollence of fuzzy sets
How to define a graded equipollence ofcountable fuzzy sets
DefinitionLet A,B ∈ Fcount. A mapping f : x → y belongs to the setBij(A,B), if
(i) f is a one-to-one mapping of x onto y,(ii) Supp(A) ⊆ x ⊆ Dom(A), and(iii) Supp(B) ⊆ y ⊆ Dom(B).
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Graded equipollence of fuzzy sets
Definition of graded equipollence betweencountable fuzzy sets
DefinitionLet A,B ∈ Fcount. We shall say that A is equipollent with B (orA has the same cardinality as B) in the degree α, if there existfuzzy sets C ∈ cls(A) and D ∈ cls(B) such that
α =∨
f∈Bij(C,D)
[C ∼�f D]
and, for each A′ ∈ cls(A), B′ ∈ cls(B) and f ∈ Bij(A′,B′), thereis [A′ ∼f B′] ≤ α.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Graded equipollence of fuzzy sets
Graded equipollence for finite fuzzy sets
TheoremLet A,B ∈ Ffin and C ∈ cls(A), D ∈ cls(B) be such that
z = Dom(C) = Dom(D) and |z| = m.
Then
[A ∼� B] =∨
f∈Perm(z)
[C ∼�f D],
where Perm(z) denotes the set of all permutations on z.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
How to model the behavior of fuzzy“cardinality” measures?
A =“Apples”
∗ ∗∗
∗
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
How to model the behavior of fuzzy“cardinality” measures?
A =“Apples”
∗ ∗∗
∗
∗
∗
∗
∗
A1
A2
A3
A4
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
How to model the behavior of fuzzy“cardinality” measures?
A =“Apples”
∗ ∗∗
∗
∗
∗
∗
∗
A1
A2
A3
A4
|A| = |A1| + |A2| + |A3| + |A4| = 4
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
How to model the behavior of fuzzy“cardinality” measures?
A =“Fresh apples”
∗ ∗∗
∗
∗
∗
∗
∗
A1 = {a/∗}
A2 = {b/∗}
A3 = {c/∗}
A4 = {d/∗}
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
How to model the behavior of fuzzy“cardinality” measures?
A =“Fresh apples”
∗ ∗∗
∗
∗
∗
∗
∗
A1 = {a/∗}
A2 = {b/∗}
A3 = {c/∗}
A4 = {d/∗}
C(Ai)(n) =?, n ∈ N
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
How to model the behavior of fuzzy“cardinality” measures?
A =“Fresh apples”
∗ ∗∗
∗
∗
∗
∗
∗
A1 = {a/∗}
A2 = {b/∗}
A3 = {c/∗}
A4 = {d/∗}
C(A) = C(A1) + C(A2) + C(A3) + C(A4)
C(Ai)(n) =?, n ∈ N
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
C(Ai)(n) =?
Crisp set Fuzzy set
C({1/∗})(0) = 0 C({a/∗})(0) = αC({1/∗})(1) = 1 C({a/∗})(1) = βC({1/∗})(2) = 0 C({a/∗})(2) = 0C({1/∗})(3) = 0 C({a/∗})(3) = 0
......
Explanation of α and β.α = “the degree of non-existence of ∗ in A1 = {a/∗}”.β = “the degree of existence of ∗ in A1 = {a/∗}”.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
C(Ai)(n) =?
Crisp set Fuzzy set
C({1/∗})(0) = 0 C({a/∗})(0) = αC({1/∗})(1) = 1 C({a/∗})(1) = βC({1/∗})(2) = 0 C({a/∗})(2) = 0C({1/∗})(3) = 0 C({a/∗})(3) = 0
......
Explanation of α and β.α = “the degree of non-existence of ∗ in A1 = {a/∗}”.β = “the degree of existence of ∗ in A1 = {a/∗}”.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Motivation
C(Ai)(n) =?
Crisp set Fuzzy set
C({1/∗})(0) = 0 C({a/∗})(0) = αC({1/∗})(1) = 1 C({a/∗})(1) = βC({1/∗})(2) = 0 C({a/∗})(2) = 0C({1/∗})(3) = 0 C({a/∗})(3) = 0
......
Explanation of α and β.α = “the degree of non-existence of ∗ in A1 = {a/∗}”.β = “the degree of existence of ∗ in A1 = {a/∗}”.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Axiomatic definition
DefinitionA class mapping C : Ffin→ N is a fuzzy c-measure of finitefuzzy sets with respect to �, if, for arbitrary A,B ∈ Ffin, it holds:
C1: if Supp(A) ∩ Supp(B) = ∅, then C(A ∪ B) = C(A) + C(B),C2: if i , j ∈ N and i > |Supp(A)|, j > |Supp(B)|, then
C(A)(i) = C(B)(j),C3: if A is a crisp set, then C(A) is a crisp set, C(A)(|A|) = >,C4: if a ∈ L and x , y ∈ Count, then C({a/x}) = C({a/y}),C5: if a,b ∈ L and x ∈ Count, then
C({a�b/x})(0) = C({a/x})(0)� C({b/x})(0),
C({a� b/x})(1) = C({a/x})(1)� C({b/x})(1).
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Axiomatic definition
ExampleConsider
Cid(A)(i) = FGCount(A)(i) =∨{a | a ∈ L and |Aa| ≥ i}
and define
C(A)(i) =
{>, i = 0,C(A)(i − 1)⊗ Cid(A)(i), otherwise.
For A = {0.5/a,0.8/b,0.1/c,0.4/d ,0/e}, we obtain
C(A) = {1/0,0.8/1,0.3/2,0/3,0/4,0/5,0/6, . . . },
where e.g. C(A)(2) = 0.8⊗ 0.5 = max(0.8 + 0.5− 1,0) = 0.3.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Axiomatic definition
Theorem (Representation of c-measures)Let C : Ffin→ N be a mapping satisfying the additivity axiomand C(A) = C(∅) for any A ∈ cls(∅). Then the followingstatements are equivalent:
(i) C is a c-measure of finite fuzzy sets with respect to �,(ii) there exist an �-homomorphism f : L→ L and an�d-homomorphism g : L→ L, such that f (⊥) ∈ {⊥,>},g(>) ∈ {⊥,>} and
C({a/x})(0) = g(a), C({a/x})(1) = f (a),
C({a/x})(k) = f (⊥), k > 1
hold for arbitrary a ∈ L and x ∈ Count.Denote Cg,f a c-measure determined by g and f .
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Axiomatic definition
Theorem (Representation of c-measures)Let C : Ffin→ N be a mapping satisfying the additivity axiomand C(A) = C(∅) for any A ∈ cls(∅). Then the followingstatements are equivalent:
(i) C is a c-measure of finite fuzzy sets with respect to �,(ii) there exist an �-homomorphism f : L→ L and an�d-homomorphism g : L→ L, such that f (⊥) ∈ {⊥,>},g(>) ∈ {⊥,>} and
C({a/x})(0) = g(a), C({a/x})(1) = f (a),
C({a/x})(k) = f (⊥), k > 1
hold for arbitrary a ∈ L and x ∈ Count.Denote Cg,f a c-measure determined by g and f .
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Expression of fuzzy c-measures
CorollaryLet L be a linearly ordered rdr-lattice, Cg,f be a c-measure suchthat f is a �-po-homomorphism and g is a�d-po-homomorphism. Then
Cg,f (A)(i) = Cg(A)(i)� Cf (A)(i)
holds for any A ∈ Ffin and i ∈ N.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Expression of fuzzy c-measures
CorollaryLet L be linearly ordered, Cg,f be a c-measure with respect to ∧such that f is ∧-homomorphism and g is ∨-homomorphisms.Then
Cg,f (A)(i) = g(Cid(A)(i + 1)) ∧ f (Cid(A)(i))
holds for any A ∈ Ffin and i ∈ N.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Example
Cg(A) for A = {0.6/x , 1/y} and g(x) = 1− x
1 2 3 4 5i
0.2
0.4
0.6
0.8
1.0
C
Cg(A)(i) = “at most i elements in A”
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Example
Cf (A) for A = {0.6/x , 1/y} and f (x) = x
1 2 3 4 5i
0.2
0.4
0.6
0.8
1.0
C
Cg(A)(i) = “at least i elements in A”
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Fuzzy c-measures of finite fuzzy sets
Example
Cg,f (A) for A = {0.6/x , 1/y} and f (x) = x ,g(x) = 1− x
1 2 3 4 5i
0.2
0.4
0.6
0.8
1.0C
Cg,f (A)(i) = “exactly i elements in A”
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Relation between graded equipollence and c-measures of finite fuzzy sets
Preliminary notions
Denotef (A) = f ◦ A
DefinitionWe shall say that fuzzy sets A and B are the equivalent fuzzysets in the degree a (symbolically, [A ≈ B] = a), if
a =∧
x∈Dom(A)∪Dom(B)
(A′(x)↔ B′(x)),
holds for A′ ∈ cls(A), B′ ∈ cls(B) with
Dom(A′) = Dom(B′) = Dom(A) ∪ Dom(B).
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Relation between graded equipollence and c-measures of finite fuzzy sets
One-to-one mappings vs. equivalence of fuzzy cardinals
TheoremLet Cg,f be a c-measure. Then
[g(A) ∼�h g(B)]� [f (A) ∼�h f (B)] ≤ [Cg,f (A) ≈ Cg,f (B)]
holds for any A,B ∈ Ffin such that |Dom(A)| = |Dom(B)| = mand h ∈ Perm(A,B).
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Relation between graded equipollence and c-measures of finite fuzzy sets
One-to-one mappings vs. equivalence of fuzzy cardinals
Corollary
Let Cg,f be a c-measure. Then(i) [g(A) ∼� g(B)] ≤ [Cg(A) ≈ Cg(B)]
(ii) [f (A) ∼� f (B)] ≤ [Cf (A) ≈ Cf (B)],hold for any A,B ∈ Ffin such that |Dom(A)| = |Dom(B)| = m.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Relation between graded equipollence and c-measures of finite fuzzy sets
One-to-one mappings vs. equivalence of fuzzy cardinals
Corollary
Let Cg,f be a c-measure. Then(i) [g(A) ∼� g(B)] ≤ [Cg(A) ≈ Cg(B)]
(ii) [f (A) ∼� f (B)] ≤ [Cf (A) ≈ Cf (B)],hold for any A,B ∈ Ffin such that |Dom(A)| = |Dom(B)| = m.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Relation between graded equipollence and c-measures of finite fuzzy sets
One-to-one mappings vs. equivalence of fuzzy cardinals
Theorem
Let L be a linearly ordered rdr-lattice, Cg,f be a c-measure suchthat f is a �-po-homomorphism and g is a�d-po-homomorphism. Then
[g(A) ∼� g(B)]� [f (A) ∼� f (B)] ≤ [Cg,f (A) ≈ Cg,f (B)]
for any A,B ∈ Ffin. Especially, if Cg and Cf are c-measures withrespect to � = ∧, then
(i) [g(A) ∼∧ g(B)] = [Cg(A) ≈ Cg(B)],(ii) [f (A) ∼∧ f (B)] = [Cf (A) ≈ Cf (B)]
hold for any A,B ∈ Ffin such that |Dom(A)| = |Dom(B)| = m.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Conclusion
A future work
To investigate further properties of fuzzy c-measures offinite fuzzy sets.To investigate further relations between fuzzy c-measuresand graded equipollence of finite fuzzy sets.To extend c-measures to infinite case.To develop the fuzzy cardinality theory.
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Relations between Graded Equipollence And Fuzzy C-measures Of Finite Fuzzy Sets
Conclusion
Thank you for your attention.